VTMO 2026 - Grade 8 - Semifinal - SET 05 - Self-study

One hundred students numbered $1$ to $100$ stand in a line. At step $k$, students whose numbers are divisible by $k$ move one step left if $k$ is odd and one step right if $k$ is even. After steps $1$ through $100$, how many students are back at their original positions?

Evaluate $2012^2-2011\cdot2013$.

Binh has ten segments of lengths $1,2,\ldots,10$. How many ways can he choose three segments that form a nondegenerate triangle?

A yellow ant starts at $B$ and a black ant at $C$ on triangle $ABC$, with $AB=20$ dm, $AC=30$ dm, $BC=40$ dm. They start together at constant speeds, the yellow ant moving $B\to A\to C\to B$ and the black ant $C\to A\to B\to C$. They first meet at $A$, then next at $M\in BC$. Find $BM$ in dm.

A box contains balls numbered $1$ through $50$. What is the least number of balls that must be drawn without looking to guarantee that two drawn numbers have a perfect-square product?

A solid regular tetrahedron has all edges $6\sqrt2$ cm. It is fully submerged in a cubic tank of side $12$ cm. When the solid is removed, by how much does the water level drop?

Isosceles right triangle $ABC$ is right at $A$, with $AB=6$ cm. Points $M\in AB$, $N\in AC$ satisfy $AM=4$ cm, $AN=2$ cm. Points $P,Q\in BC$ are chosen so that $PM$ and $QN$ are perpendicular to $MN$. Find the area of quadrilateral $MNQP$ in cm².

How many integers $x$ satisfy $(x^2-1)(x^2-10)(x^2-20)<0$?

For a positive integer $n$, the units digit of $n^3+3n^2+3n$ is $4$. Find the units digit of $31n^2+n+2021$.

A reduced fraction $a/b$ has the property that adding $36$ to both its numerator and denominator gives half its original value. Find $a+b$.

Let $A=1^3+2^3+\cdots+12^3$. How many of these statements are true: (1) $A$ is divisible by $12$ but not $13$; (2) divisible by $13$ but not $12$; (3) a perfect square; (4) divisible by $156$; (5) divisible by $169$?

Dice are placed side by side in one straight row on a table. All exposed faces are counted from every direction: two dice expose $8$ faces, three expose $11$. If $50$ faces are exposed, how many dice are there?

Twenty-five giraffes all have different heights. In each round exactly five can be compared and ranked by height. Describe how to determine the tallest, second tallest and third tallest giraffes in seven rounds.

Solve $\dfrac{x-1}{x}-\dfrac{x}{x-1}=0$.

In the coordinate plane, rectangle $OMNP$ has vertices $O=(0,0)$, $M=(0,5)$, $N=(10,5)$, $P=(10,0)$. How many integer-coordinate points $(x,y)$ lie strictly inside it and satisfy $x+y<10$?

The internal angle bisectors of triangle $ABC$ meet at $I$. The angles $\angle BIC,\angle CIA,\angle AIB$ have ratio $8:9:13$. Find $\angle A$ in degrees.

Solve $\dfrac{x+1}{x-2}+\dfrac{3-2x}{2-x}=0$.

An, Binh and Chau make two statements each about Dung\'s age. An: “He is not $10$. He is $11$.” Binh: “He is not $11$. He is not $9$.” Chau: “He is not $12$. He is $10$.” One friend has both statements true, one has exactly one true, and one has both false. How old is Dung?

Nam makes four claims about a number $A$: $A-4$ is prime; $A-5$ is a perfect square; $4A-3$ is a perfect square; and $4A+12$ has exactly six positive divisors. Exactly one claim is false. Find $A$.

Given $4^a=5$, $5^b=6$, $6^c=7$, $7^d=8$, find $abcd$.